This question comes from some thoughts I had after reading this question.
First of all, is an orbital an observable? I know the answer to this question is no because there is no "single-orbital operator" or whatever you'd like to call this. I brig this up because exactly what this means should be a part of a thorough answer. Plus an orbital is a state and we don't observe the eigenvectors but the eigenvalues.
Second, is the energy of an orbital an observable? It's more tempting to think that the answer to this is yes, but if it's true that there is no "single-orbital operator" then there must be no corresponding eigenvalue which represents the energy of this orbital. As I understand it, if one runs a HF calculation and sums the energies calculated for the individual orbitals, this is the total electronic energy. In the answer linked above, however, it is said that there is an infinite number of unitary transformations for a given set of orbitals to another set. Is there any correspondence between the energies of these transformed orbitals and the original set of orbitals? By that I mean, the orbitals themselves change, but would the optimized energies of each orbital change and it is the only the wavefunction and total energy which stay constant?
Finally, how do the answers to the above questions connect with Koopmans' theorem and photoelectron spectroscopy? That is to say, Koopmans' theorem says that the energy of the HOMO as calculated from Hartree-Fock corresponds to the first ionization energy of the system. Additionally, in photoelectron spectroscopy, it sure seems like people are observing the energies of individual orbitals, but I don't really know anything about this technique in detail so that may just be a misunderstanding.
Additionally, is Koopmans' theorem still true once we perform a unitary transformation on the Hartree-Fock wavefunction?
The reason I lump all these questions together is because I think in order to sufficiently answer the question of whether or not an orbital energy is observable, all of these points should be addressed, so I just asked them all at once. I hope it's not overkill.
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